So, given that constraint, here is how I might approach it:
In order for the Promise resolve function to fire, you need a 2xx response status code. So, in LoginController, override teh logout method that is defined in the AuthenticatesUsers trait:
// LoginController
public function logout(Request $request)
{
$this->guard()->logout();
$request->session()->invalidate();
if ($request->wantsJson()) {
return response()->json([], 204);
}
return redirect('/');
}
Now, you can respond with an empty (but successful) JSON response, which will allow you to define how/where to relocate in the Vue app. I don't think location.reload() is appropriate since you will probably be on a page that required authentication before you logged out, so if using Vue Router:
router.push('login')
or, if not
document.location.href = "/login";